Many wireless communication systems transfer information by modulating the information onto a carrier signal such as a sine wave to more efficiently use the available bandwidth for multiple or intensive communications. The carrier signal is modulated by varying one or more parameters such as amplitude, frequency, and phase. Phase shift keying (PSK) is frequently used, and includes shifting the phase of the carrier according to the content of the information being transmitted. There are several techniques known to modulate a transmission phase including binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), and Guassian minimum shift keying (GMSK). The power spectral density of both BPSK and QPSK is fairly broad, and these techniques have been found inadequate for certain applications due to interference between closely packed adjacent channels. A signal transmitted with phase shift keying must be demodulated at the receive end of a communication system by estimating the phase shift or offset. Data demodulation is directly dependent upon the accuracy of that estimate of the phase shift.
GMSK is a form of continuous phase modulation, and therefore achieves smooth phase modulation that requires less bandwidth than other techniques. Under GMSK, input bits defining a rectangular waveform (+1, −1) are converted to Gaussian (bell shaped) pulses by a Gaussian smoothing filter. The Gaussian pulse typically is allowed to last longer than its corresponding rectangular pulse, resulting in pulse overlap known as intersymbol interference (ISI). The extent of ISI is determined by the product of the bandwidth (B) of the Gaussian filter and the data-bit duration (Tb) or bit rate; the smaller the product, the greater the pulse overlap. Applications using GMSK have been used where the product BTb is generally 0.3 or greater. Applications with lower BTb (e.g., ⅕, ⅙, and less) tend to include higher levels of ISI that generally degrade performance to unacceptable levels. This is true because prior art demodulators introduce a phase error (the difference between the actual phase offset in the transmitted signal and the estimate of the phase offset in the received signal) that increases with decreasing BTb.
In a burst transmission system, an estimate of the phase offset is typically included within a header or training sequence of the message stream. Theoretically, where the length L0 of the training sequence approaches infinity, the estimate of the phase offset exactly replicates the actual phase offset and phase error approaches zero. In more pragmatic applications, the use of finite length training sequences results in a discrepancy between the estimate of the phase offset and the actual phase offset. The maximum allowable size of this discrepancy depends upon the demands of a particular communications system. The required training sequence length increases as the maximum allowable discrepancy decreases and as BTb decreases. Because the training sequence header is present in each transmission burst, shorter training sequences are desirable to ensure that that the available bandwidth is used for substantive data rather than inordinately long training sequences.
Phase error in prior art systems arises during demodulation. While a GMSK waveform defines a constant envelope that is simple to generate and transmit using efficient amplifiers, it is a fairly complex function to demodulate at the receiving end of a communication system.
Pierre Laurent first demonstrated that a GMSK waveform could be represented by amplitude modulated pulses h0(t), h1(t), h2(t), . . . , and a corresponding coherent detector could be designed. This representation is known as the Laurent decomposition, and it allowed bit error rates (BER) associated with GMSK communications to match those of other PSK techniques. Prior art has demonstrated that detectors based on only the first two amplitude pulses, h0(t) and h1(t), or only the first amplitude pulse h0(t), provide adequate performance for most applications.
Many GMSK systems calculate the phase estimate as the angle resulting from the inner product of two vectors, one representing pseudo symbols (which are related to channel symbols and the modulation index) and the other representing the filter output (which is matched to the first Laurent pulse). The above approach yields only an approximation since the last L filter outputs, x(L0−1−L), x(L0−L), . . . , x(L0−1), are integrated over less than a full symbol, because the first Laurent pulse spans (L+1)Tb seconds. This is the bias in demodulating a GMSK signal that has previously limited GMSK systems to about BTb>0.3 and high L0. At those system parameters, the above bias in the Laurent decomposition can be ignored without adverse effect on demodulation at the receive end of the communication. At lower BTb using shorter L0, the bias becomes more significant and cannot be ignored; the bias signal drives the phase estimate further from the true phase modulation, causing the BER to rise.